The Weierstrass half‐periods and the Weierstrass invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are defined by the following formulas:

where is the Klein invariant modular function, is Weierstrass elliptic function, and is theWeierstrass zeta function.

The previous four vector functions are sometimes called the Weierstrass utilities because they are the basic elements of the Weierstrass theory of elliptic functions.